IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011 75
Leaderless and Leader-Following Consensus WithCommunication and Input Delays Under
a Directed Network TopologyZiyang Meng, Wei Ren, Member, IEEE, Yongcan Cao, Student Member, IEEE, and Zheng You
Abstract—In this paper, time-domain (Lyapunov theorems) andfrequency-domain (the Nyquist stability criterion) approaches areused to study leaderless and leader-following consensus algorithmswith communication and input delays under a directed networktopology. We consider both the first-order and second-order casesand present stability or boundedness conditions. Several interest-ing phenomena are analyzed and explained. Simulation results arepresented to support the theoretical results.
Index Terms—Communication and input delays, consensustracking, directed network graph, leaderless consensus, multi-agent system.
I. INTRODUCTION
COOPERATIVE control of multiagent systems has re-ceived significant research attention in recent years. Com-
pared with solo systems, additional benefits, such as highrobustness and great efficiency, can be obtained by having agroup of agents work cooperatively. Cooperative control hasbroad applications in formation control [1], flocking [2], andcomplex networks [3], [4]. A fundamental approach to achievecooperative control is consensus [5]–[7]. Consensus meansthe agreement of a group of agents on their common statesvia local interaction. In a leaderless consensus problem, theredoes not exist a virtual leader, while in a leader-followingconsensus problem, there exists a virtual leader that specifiesthe objective for the whole group. More specifically, consensuswith a static virtual leader is called a consensus regulationproblem, and consensus with a dynamic virtual leader is calleda consensus tracking problem. It is worthwhile to mention thatthe consensus tracking problem becomes much more complexif only a portion of the agents in the group has access to thevirtual leader.
Since delays are inevitable in real systems, it is necessary andbeneficial to study leaderless and leader-following consensusalgorithms in the presence of the delays. Most existing refer-
Manuscript received July 23, 2009; revised December 28, 2009; acceptedMarch 6, 2010. Date of publication April 29, 2010; date of current versionJanuary 14, 2011. This work was supported by a National Science FoundationCAREER Award (ECCS-0748287). The work of Z. Meng was supported in partby the Ministry of Education of China and in part by China Scholarship Councilunder Grant LiuJinChu[2008]3019-2008621094. This paper was recommendedby Associate Editor E. Tunstel.
Z. Meng and Z. You are with the Department of Precision Instruments andMechanology, Tsinghua University, Beijing 100084, China.
W. Ren and Y. Cao are with the Department of Electrical and Com-puter Engineering, Utah State University, Logan, UT 84322 USA (e-mail:[email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSMCB.2010.2045891
ences on consensus algorithms considered input delays. Theauthors in [6] first gave a leaderless consensus algorithm withinput delays and then presented a frequency-domain approachto find the stability conditions. A similar leaderless consensusalgorithm with uniform input delays was studied in [8], wherea time-domain approach, i.e., the Lyapunov–Krasovaskii theo-rem, was used to obtain the stability conditions under stronglyconnected and balanced network topologies. Besides leaderlessconsensus algorithms, leader-following consensus algorithmswith input delays were also studied. By combining the resultsin [8] and [9], the authors in [10] proposed a first-order consen-sus tracking algorithm with input delays, where an estimatorwas used to estimate the virtual leader’s velocity. Due to thepresence of the dynamic virtual leader and the input delays, thetracking errors were shown to be uniformly ultimately boundedinstead of approaching zero. In the previous references, thenetwork topology is assumed to be either undirected or stronglyconnected and balanced, which poses an obvious limitation.The extension to the case where the network topology has adirected spanning tree and the input delays are assumed to benonuniform was provided in [11], where a frequency-domainmethod was used to find conditions to achieve leaderless con-sensus. Except for input delays, the influence of communicationdelays on consensus algorithms was also studied. The authors in[12] showed that communication delays will not jeopardize thestability of the first-order leaderless consensus algorithm undera directed network topology. A similar algorithm was discussedin [13], where the effect of initial conditions was highlighted. Asecond-order consensus regulation algorithm with nonuniformcommunication delays was studied in [14], but a dampingterm was used to regulate the velocities of all agents to zero,and the network topology was assumed to be undirected. Theprevious references considered either only the input delays oronly the communication delays and, hence, lack completeness.The case with both the communication and input delays wasstudied in [15]. In particular, a first-order leaderless consensusalgorithm with both the communication and input delays wasstudied in a discrete-time setting. However, a pure frequency-domain approach was used, thus leading the obtained stabilityconditions to be conservative.
This paper considers both leaderless and leader-followingconsensus algorithms with communication and input delays in,respectively, first-order kinematics and second-order dynamicsunder a directed network topology. The stability or bound-edness conditions of four different cases, namely, leaderlessconsensus, consensus regulation, consensus tracking with fullaccess to the virtual leader, and consensus tracking with partial
76 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
access to the virtual leader, are analyzed by using time-domainand frequency-domain approaches.
The contributions of this paper are fourfold. First, we assumea general network topology, i.e., a directed network topologywith a directed spanning tree, instead of an undirected con-nected network topology or a directed strongly connected andbalanced network topology [6], [8], [10], [16]. Second, bothcommunication and input delays are considered in the casesof leaderless consensus, consensus regulation, and consensustracking with full access to the virtual leader, which guaranteesthe completeness of the algorithms. Third, we show that thecommunication delay will not influence the stability of the first-order system in the case of consensus tracking with partialaccess to the virtual leader, which extends the results of [12]and [17]. Fourth, as a byproduct, we find that in the case ofsecond-order leaderless consensus with both communicationand input delays, the final group velocity is always dampened tozero rather than a possibly nonzero constant as compared withthe standard second-order consensus algorithm studied in [18].A preliminary version of the second-order case of the work ispresented at the 2010 American Control Conference.
II. PRELIMINARIES
A. Notations
R and C are the set of real numbers and the set of complexnumbers, respectively.
1n and 0n are the n × 1 all-one vector and the n × 1 all-zerovector, respectively.
In and 0n×n are the n × n identity matrix and the n × nmatrix with all zero entries, respectively.
λmin(A) and λmax(A) are, respectively, the minimal eigen-value and the maximum eigenvalue of the matrix A.‖A‖ is the norm of the matrix/vector A.j is the imaginary unit.�{•} and �{•} are, respectively, the real part and the
imaginary part of a complex number.ρ(A) is the spectral radius of the matrix A.Q < 0 means that the matrix Q is negative-definite.
B. Graph Theory Notions
Using graph theory, we can model the network topology in amultiagent system consisting of n agents. A directed graph Gn
consists of a pair (V, E), where V = {v1, . . . , vn} is a finitenonempty set of nodes, and E ∈ V × V is a set of ordered pairsof nodes. An edge (vi, vj) denotes that node vj can obtaininformation from node vi, but not necessarily vice versa. Allneighbors of node vi are denoted as Ni := {vj | (vj , vi) ∈ E}.
A directed path is a sequence of edges of the form(vi1 , vi2), (vi2 , vi3), . . .. A directed graph has a directed span-ning tree if there exists at least one node having a directed pathto all other nodes.
For the leaderless consensus case, the adjacency matrixAn = [aij ] ∈ Rn×n associated with Gn is defined such thataij is positive if (vj , vi) ∈ E, while aij = 0 otherwise. Here,we assume that aii = 0, ∀ i. The (nonsymmetric) Laplacianmatrix Ln = [�ij ] ∈ Rn×n associated with An is defined as�ii =
∑j �=i aij and �ij = −aij , where i �= j.
For the leader-following case, we assume that besides agents1 to n, there exists a virtual leader, labeled as agent n + 1, in
the system. We use Gn+1 to model the network topology inthis case. The adjacency matrix An+1 = [aij ] ∈ R(n+1)×(n+1)
associated with Gn+1 is defined such that aij is positive if(vj , vi) ∈ E, while aij = 0 otherwise, and a(n+1)j = 0 for allj = 1, . . . , n + 1. Here, again, we assume that aii = 0, ∀ i.
III. DEFINITIONS AND LEMMAS
Suppose that f : R × C �→ Rn is continuous and considerthe retarded functional differential equation (RFDE)
x(t) = f(t, xt). (1)
Let φ = xt be defined as xt(θ) = x(t + θ), θ ∈ [−τ, 0]. Sup-pose that appropriate initial conditions are defined on the delayinterval [t0 − τ, t0]: xt0(θ) = φ(θ), ∀ θ ∈ [−τ, 0]. Specifically,we assume that the initial condition satisfies x(θ) = 0, ∀ θ ∈[t0 − τ, t0], in this paper. Suppose that the solution x(σ, φ)(t)through (σ, φ) is continuous in (σ, φ, t) in the domain ofdefinition of the function, where σ ∈ R.
Definition 3.1 [19]: The solutions x(σ, φ) of the RFDE(1) are uniformly ultimately bounded if there is a β > 0such that for any α > 0, there is a constant t0(α) > 0 suchthat |x(σ, φ)(t)| ≤ β for t ≥ σ + t0(α) for all σ ∈ R, φ ∈ C,|φ| ≤ α.
Suppose that D : R × C �→ Rn is a linear operator onthe second variable such that D(t, φ) = A(t)φ(0) − G(t, φ),where A(t) is a continuous nonsingular matrix, and G(t, φ) =∫ 0
−h dμ(t, θ)φ(θ) satisfies |∫ 0
−s+ [dμ(t, θ)]φ(θ)| ≤ γ(s, t)|φ|for 0 ≤ s ≤ h, where μ is an n × n matrix function of boundedvariation on θ, γ is continuous, and γ(0, t) = 0 for t ≥ 0. Ifg : R × C �→ Rn is a continuous function, then the relation
d
dtD(t, xt) = g(t, xt) (2)
is a neutral functional differential equation (NFDE) [20].Definition 3.2 [20]: Consider the NFDE (2). Suppose that
operator D is stable. It defines a uniform ultimately boundedprocess if there is a β > 0 such that for any α > 0, there isa constant t0(α) > 0 such that |x(σ, φ)(t)| ≤ β for t ≥ σ +t0(α) for all σ ∈ R, φ ∈ C, |φ| ≤ α.
Lemma 3.1. (Degenerate Lyapunov–Krasovskii StabilityTheorem) [21], [22]: Consider the NFDE (2). Suppose thatoperator D is stable, g : R × C �→ Rn takes R× (bounded setsof C) into bounded sets of Rn, and u(s), v(s), and w(s)are continuous, nonnegative, and nondecreasing functions withu(s), v(s) > 0 for s �= 0 and u(0) = v(0) = 0. If there exists acontinuous functional V : R × Cn × Cn �→ Rn, such that
1) u(‖D(t, φ)‖) ≤ V (t,D(t, φ), φ) ≤ v(‖φ‖c)2) V (t,D(t, φ), φ) ≤ −w(‖D(t, φ)‖)then the trivial solution of (2) is asymptotically stable.Lemma 3.1 will be used in the first-order and second-order
leaderless consensus and consensus regulation problems.Lemma 3.2. (Lyapunov–Razumikhin Uniformly Ultimately
Bounded Theorem) [19]: Consider the RFDE (1). Supposethat f : R × C �→ Rn takes R× (bounded sets of C) intobounded sets of Rn and u, v, w : R+ �→ R+ are continuousnonincreasing functions, u(s) → ∞ as s → ∞. If there is acontinuous function V : R × Rn �→ R, a continuous nonde-creasing function p : R+ �→ R+, p(s) > s for s > 0, and a con-stant H ≥ 0 such that u(|x|) ≤ V (x) ≤ v(|x|), t ∈ R, x ∈ Rn,
MENG et al.: LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS 77
and V (t, φ) ≤ −w(|φ(0)|) if |φ(0)| ≥ H , V (t + θ, φ(θ)) <p(V (t, φ(0))), θ ∈ [−τ, 0], then the solutions of (1) are uni-formly ultimately bounded.
Lemma 3.2 will be used in the first-order and second-orderconsensus tracking problems with full access to the virtualleader.
Lemma 3.3. (Lyapunov–Razumikhin Uniformly UltimatelyBounded Theorem for Neutral-Type Systems) [19], [20]: Con-sider the NFDE (2). Suppose that operator D is stable andg : R × C �→ Rn takes R× (bounded sets of C) into boundedsets of Rn. If there is a continuous function V : R × Rn �→ R,a continuous nondecreasing function p : R+ �→ R+, p(s) > sfor s > 0 such that u(|x|) ≤ V (x) ≤ v(|x|), ∀x ∈ Rn, andV (D(t, φ)) ≤ −w(|D(t, φ)|) for all functions φ if |D(t, φ)| ≥H and V (φ(θ)) < p(V (D(t, φ))), θ ∈ [−τ, 0], where w(s) is acontinuous positive function for s ≥ KH , then the solution of(2) is uniformly ultimately bounded.
Lemma 3.3 will be used in the first-order and second-orderconsensus tracking problems with partial access to the virtualleader.
IV. FIRST-ORDER CASE WITH COMMUNICATION AND
INPUT DELAYS UNDER A DIRECTED NETWORK TOPOLOGY
Here, we model a group of agents with single-integratorkinematics as
xi(t) = ui(t), i = 1, 2, . . . , n (3)
where xi and ui are, respectively, the state and the control inputof the ith agent.
A. First-Order Leaderless Consensus
Consider the following leaderless consensus algorithm withboth communication and input delays:
ui(t) = − 1∑nj=1 aij
n∑j=1
aij [(xi(t − τ1) − xj(t − τ1 − τ2)] ,
i = 1, . . . , n, (4)
where τ1 and τ2 are the input and communication delays,respectively, and aij , i = 1, . . . , n, j = 1, . . . , n, is the (i, j)entry of the adjacency matrix An. Here, we assume that everyagent has a neighbor, which implies that
∑nj=1 aij �= 0, ∀ i. To
achieve consensus, that is, xi(t) → xj(t), as t → ∞, the con-ditions on the input delay τ1 and the communication delay τ2 toguarantee the stability or the ultimately uniform boundednessof the closed-loop system should be addressed. Using (4), (3)can be written in the matrix form as
x(t) = −x(t − τ1) + Ax(t − τ1 − τ2) (5)
where x = [x1, . . . , xn]T , and A = [aij ] ∈ Rn×n is defined asaij = aij/
∑nj=1 aij , i = 1, . . . , n, j = 1, . . . , n. Define L =
In − A. When Gn has a directed spanning tree, L has asimple zero eigenvalue, and all other eigenvalues are on theopen right half-plane [7], [23]. The following singular vectordecomposition is valid:
W−1LW =[
L 0n−1
0Tn−1 0
].
Here, among the infinite options of W , we choose the one thatthe last column of W is the vector 1n. Note that, here, allthe eigenvalues of L are on the open right half-plane. Beforemoving on, we need the following lemma.
Lemma 4.1 [10]: For any a, b ∈ Rn and any symmetricpositive-definite matrix Φ ∈ Rn×n, 2aT b ≤ aT Φ−1a + bT Φb.
Define xΔ= W−1x. Denote xn−1 as the first n − 1 rows of
x and x2 as the last row of x. Note that A = In − L. Bymultiplying W−1 on both sides of (5), it follows that (5) canbe rewritten as[ ˙xn−1(t)
˙x2(t)
]= −
[In−1 0n−1
0Tn−1 1
] [xn−1(t − τ1)x2(t − τ1)
]
+[
A 0n−1
0Tn−1 1
] [xn−1(t − τ1 − τ2)x2(t − τ1 − τ2)
]
where A = In−1 − L. Equation (5) can be decoupled into thefollowing two equations:
Theorem 4.1: If the fixed directed graph Gn has a directedspanning tree and every agent has a neighbor, there exist τ1 andτ2 such that the following three conditions1 are satisfied.
A)] < 0, where Pfc is a symmetric positive-definitematrix chosen properly such that (−In−1 + A)T Pfc +Pfc(−In−1 + A) < 0, and Sfc and Hfc are arbitrarysymmetric positive-definite matrices.
In addition, if the above conditions are satisfied, τ1 ∈ [0, τ1],and τ2 ∈ [0, τ2], system (3) using (4) reaches the consensusequilibrium (pT x(0)/(1 + τ2))1n asymptotically, where p ∈Rn is a nonnegative left eigenvector of L associated with thezero eigenvalue satisfying pT 1n = 1.
Proof: We first prove that the stability of system (3) using(4) is guaranteed if the three conditions in Theorem 4.1 aresatisfied. Then, we show that these three conditions are, indeed,satisfied if Gn has a directed spanning tree, and every agenthas a neighbor. At last, the consensus equilibrium is explicitlypresented by using the final value theorem.
We know that the stability of the following system:
d
dt
⎛⎝xn−1(t) −
0∫−τ1
xn−1(t + θ) dθ + A
0∫−τ1−τ2
xn−1(t + θ) dθ
⎞⎠
= −(In−1 − A)xn−1(t) (7)
1Note here that the three conditions are used to obtain the upper bounds τ1and τ2 for allowable delays.
78 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
implies the stability of system (6a) if condition 2 inTheorem 4.1 is satisfied [22]. Consider a Lyapunov functioncandidate
V(x(n−1)t
)=
⎡⎣xn−1(t) −
0∫−τ1
xn−1(t + θ) dθ
+ A
0∫−τ1−τ2
xn−1(t + θ) dθ
⎤⎦
T
× Pfc
⎡⎣xn−1(t) −
0∫−τ1
xn−1(t + θ) dθ
+ A
0∫−τ1−τ2
xn−1(t + θ) dθ
⎤⎦
+
0∫−τ1
t∫t+θ
xn−1(ξ)T Sfcxn−1(ξ) dξ dθ
+
0∫−τ1−τ2
t∫t+θ
xn−1(ξ)T Hfcxn−1(ξ) dξ dθ.
Taking the derivative of V along (7) gives
V(x(n−1)t
)≤ xn−1(t)T Qfcxn−1(t)
where Qfc is defined as in Theorem 4.1, and we have usedLemma 4.1 to derive the inequality. Note that Qfc < 0 satisfiescondition 2 in Lemma 3.1. Also, note that α1‖D(x(n−1)t)‖ ≤V(x(n−1)t)≤α2‖x(n−1)t‖c [24], where D(x(n−1)t)= xn−1(t)−∫ 0
−τ1xn−1(t + θ) dθ+A
∫ 0
−τ1−τ2xn−1(t + θ) dθ, ‖x(n−1)t‖c =
supθ∈[−τ1−τ2,0] ‖x(n−1)(t + θ)‖, α1 = λmin(Pfc), and α2 =λmax(Pfc) + τ1λmax(Sfc) + (τ1 + τ2)λmax(Hfc). This satis-fies condition 1 in Lemma 3.1. Therefore, if conditions 2 and3 in Theorem 5.1 are satisfied, the asymptotical stability ofsystem (6a) is guaranteed by using Lemma 3.1.
For system (6b), we apply the Nyquist stability criterionto find its stability condition. After Laplace transformation,system (6b) can be written as
sx2(s) = −e−τ1sx2(s) + e−(τ1+τ2)sx2(s).
Thus, the stability is determined by the roots’ distribution of thefollowing:
s = −e−τ1s + e−(τ1+τ2)s. (8)
Define f(s) Δ= (e−τ1s − e−(τ1+τ2)s)/s. Based on the Nyquiststability criterion, if the trajectory of f(jω), ∀ω ∈ (−∞,∞),does not enclose the point (−1, j0), then (8) is stable. Onesufficient condition is that �{f(jω)} > −1, ∀ω ∈ (−∞,∞).Noting that �{f(jω)} = (− sin(τ1 + τ2)ω/ω) + (sin τ1ω/ω)and functions (− sin(τ1 + τ2)ω/ω) and (sin τ1ω/ω) have min-imum values, respectively, −(τ1 + τ2) and −τ1 with respectto ∀ω ∈ (−∞,∞), we have that �{f(jω)} ≥ −(2τ1 + τ2).Therefore, it is easy to verify that the stability of system (6b)is guaranteed if condition 1 in Theorem 4.1 is satisfied.
Next, we show that these three conditions in Theorem 4.1are, indeed, satisfied if Gn has a directed spanning tree, and
every agent has a neighbor. It is straightforward to see that thereexist τ1 and τ2 such that conditions 1 and 2 are satisfied. Forcondition 3, we know that L = In−1 − A has all eigenvalues onthe open right half-plane. Therefore, when τ1 = τ2 = 0, therealways exists a Pfc to guarantee that (−In−1 + A)T Pfc +Pfc(−In−1 + A) < 0. Thus, based on the continuity, theremust exist τ1 and τ2 such that Qfc < 0 when τ1 ∈ [0, τ1] andτ2 ∈ [0, τ2].
Finally, for the consensus equilibrium, we have that
limt→∞
x2(t) = lims→0
sx2(0)s + e−τ1s − e−(τ1+τ2)s
=x2(0)1 + τ2
and xn−1(t) → 0, as t → ∞. It follows that the consensusequilibrium is given by pT x(0)/(1 + τ2)1n. �
Remark 4.1: We know that the additional dynamics causedby the model transformation from (6a) to (7) can be character-ized by the solutions of the following complex equation [25]:
det(
In−1 − In−11 − esτ1
s+ A
1 − es(τ1+τ2)
s
)= 0, s ∈ C.
Thus, if τ1 + (τ1 + τ2)‖A‖ < 1, there are no additional eigen-values induced by the model transformation from (6a) to (7),which implies that the condition τ1 + (τ1 + τ2)‖A‖ < 1 can beused to replace condition 2 in Theorem 4.1.
Remark 4.2: If we let Sfc = Hfc = In−1, condition 3 inTheorem 4.1 can be written as
τ1 + τ2 <λmin
[(In−1−A)T Pfc+Pfc(In−1−A)
]2+
∥∥∥(−In−1+A)T Pfc
∥∥∥2
+∥∥∥(−In−1+A)T PfcA
∥∥∥2 .
Remark 4.3: Note that, in Theorem 4.1, it is assumed thatthe fixed directed graph has a directed spanning tree, and everyagent has a neighbor. Thus, the conclusion can be viewed as ageneralization of [8], [10], and [16], where the directed graphsare assumed to be strongly connected and balanced.
Remark 4.4: For first-order leaderless consensus, the caseof a general network topology that has a directed spanningtree was also considered in [11]. However, only input delayswere considered. The extension to the case where there existboth communication and input delays was studied in [15].A discrete-time setting was assumed, and a pure frequency-domain approach was used. In contrast, we here introduceboth time-domain and frequency-domain approaches in acontinuous-time setting.
B. First-Order Consensus Regulation
Here, we assume that there exists a virtual leader, labeled asagent n + 1, whose state is a constant reference state xd. Theconsensus regulation algorithm with both communication andinput delays is proposed as
ui = − 1∑n+1j=1 aij
n+1∑j=1
aij [xi(t − τ1) − xj(t − τ1 − τ2)] ,
i = 1, . . . , n (9)
where τ1 and τ2 are, respectively, the input and communicationdelays, aij , i = 1, . . . , n, j = 1, . . . , n + 1, is the (i, j) entry
MENG et al.: LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS 79
of the adjacency matrix An+1, and x(n+1) ≡ xd. Note thatthe condition that Gn+1 has a directed spanning tree and thefact that all entries of the last row of An+1 are zero implythat no other rows of An+1 have all zero entries. It thusfollows that
∑n+1j=1 aij �= 0, i = 1, 2, . . . , n [26]. The objective
of (9) is to guarantee accurate regulation, i.e., xi(t) → xd ast → ∞. Denote xi = xi − xd and x = [x1, . . . , xn]T . DefineA = [aij ] ∈ Rn×n as aij = aij/
∑n+1j=1 aij , i = 1, . . . , n, j =
1, . . . , n. Using (9), (3) can be written in the matrix form as
x = −x(t − τ1) + Ax(t − τ1 − τ2) (10)
where we have used the fact that xd is a constant. Beforemoving on, we need the following lemma regarding (In −A).
Lemma 4.2 [27]: The real parts of all eigenvalues of (In −A) are positive if the fixed directed graph Gn+1 has a directedspanning tree.
Theorem 4.2: If the fixed directed graph Gn+1 has a directedspanning tree, there exist τ1 and τ2 such that the following twoconditions are satisfied.
where Pfr is a symmetric positive-definite matrix chosenproperly such that (−In + A)T Pfr + Pfr(−In + A) <0, and Sfr and Hfr are arbitrary symmetric positive-definite matrices.
In addition, if the above conditions are satisfied, τ1 ∈ [0, τ1],and τ2 ∈ [0, τ2], system (3) using (9) guarantees xi(t) → xd,∀ i = 1, . . . , n, asymptotically as t → ∞.
Proof: Similar to the analysis in Section IV-A, the stabil-ity of the following system:
d
dt
⎛⎝x(t) −
0∫−τ1
x(t + θ) dθ + A0∫
−τ1−τ2
x(t + θ) dθ
⎞⎠
= −(In −A)x(t) (11)
implies the stability of system (10) if condition 1 inTheorem 4.2 is satisfied.
Consider a Lyapunov function candidate
V (xt) =
⎡⎣x(t)−
0∫−τ1
x(t+θ) dθ + A0∫
−τ1−τ2
x(t+θ) dθ
⎤⎦
T
×Pfr
⎡⎣x(t)−
0∫−τ1
x(t+θ) dθ + A0∫
−τ1−τ2
x(t+θ) dθ
⎤⎦
+
0∫−τ1
t∫t+θ
x(ξ)T Sfrx(ξ) dξ dθ
+
0∫−τ1−τ2
t∫t+θ
x(ξ)T Hfrx(ξ) dξ dθ.
Taking the derivative of V along (11) gives
V (xt) ≤ x(t)T Qfrx(t)
where Qfr is defined as in Theorem 4.2, and we have usedLemma 4.1 to derive the inequality. Thus, if the two conditionsin Theorem 4.2 are satisfied, the stability of (10) can be guar-anteed by using Lemma 3.1. In addition, it is straightforwardto see that there exist τ1 and τ2 such that condition 1 issatisfied. For condition 2, we know that there also exist τ1 andτ2 such that Qfr < 0 by following a similar analysis to thatin Section IV-A since In −A has all eigenvalues with positivereal parts if Gn+1 has a directed spanning tree (Lemma 4.2). �
Remark 4.5: Although the approaches used in the leaderlessconsensus case and the consensus regulation case are similar,the control goals of these two cases are different. For the leader-less consensus case, the final states of each agent are determinedby the network topology, the control gains, and the time delaysrather than being prespecified. However, for the consensusregulation case, there exists a virtual leader that determines thefinal state, and the control objective is to guarantee that the finalstates of all agents approach the state of the virtual leader. Plus,the result of the case of consensus regulation can be generalizedto general weights, while the case of leaderless consensusrequires special weights. Also, note that the remarks given inRemarks 4.1–4.3 are still valid in the consensus regulation case.
Remark 4.6: Using the similar model and analysis providedin [28], the results in this subsection can be extended to the caseof multiple (nonuniform) delays.
C. First-Order Consensus Tracking With Full Access to theVirtual Leader
Here, we consider the case where the reference state xd istime varying. Here, we assume that all agents have access toxd. The consensus tracking algorithm with both communicationand input delays is proposed as
ui = xd(t − τ1 − τ2) −1∑n+1
j=1 aij
n+1∑j=1
aij [xi(t − τ1)
− xj(t − τ1 − τ2)] , i = 1, . . . , n (12)
where τ1 and τ2 are, respectively, the input and communicationdelays, aij , i = 1, . . . , n, j = 1, . . . , n + 1, is the (i, j) entryof the adjacency matrix An+1, and xn+1 ≡ xd. Using (12), (3)can be written in the matrix form as
x = −x(t − τ1) + Ax(t − τ1 − τ2) + Rft (13)
where A and x are defined as in Section IV-B, and Rft =1n[xd (t−τ1−τ2)−xd (t)]−1n[xd (t−τ1)−xd (t−τ1−τ2)]=−1n
∫ 0
−τ1−τ2xd(t + θ) dθ − 1n
∫ 0
−τ2xd(t + θ) dθ by using the
Leibniz–Newton formula [19]. Here, we also assume that |xd|<δv, |xd| < δa, where δv and δa are two positive constants.
Theorem 4.3: If the fixed directed graph Gn+1 has adirected spanning tree, there exist τ1 and τ2 such that Qft =(−In+ A)TPfr+ Pfr(−In+ A)+ τ1(Pfr+ PfrAP−1
frATPfr+2qfPfr)+ (τ1 + τ2)(PfrAP−1
frATPfr+ PfrAAP−1frATATPfr+
2qfPfr) < 0, where Pfr is the same matrix given inTheorem 4.2 and qf > 1. In addition, if Qft < 0, τ1 ∈ [0, τ1],and τ2 ∈ [0, τ2], system (3) using (12) guarantees that allxi − xd are uniformly ultimately bounded. In particular,the ultimate bound of x is given by λmax(Pfr)af/λmin(Pfr)κfλmin(−Qft), where af = 2[(τ1 + τ2)δa +τ2δv][‖Pfr‖ + τ1‖Pfr‖ + (τ1 + τ2)‖PfrA‖] and 0 < κf < 1.
80 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
Proof: Using the Leibniz–Newton formula [19], we trans-form (13) to the following system:
d
dtx(t) = − (In −A)x(t) +
0∫−τ1
x(t + θ) dθ
−A0∫
−τ1−τ2
x(t + θ) dθ + Rft
= − (In −A)x(t)
+
0∫−τ1
[−x(t − τ1 + θ) + Ax(t − τ1 − τ2 + θ)] dθ
+
0∫−τ1
Rft(t + θ) dθ
−A0∫
−τ1−τ2
[−x(t−τ1+θ) +Ax(t−τ1−τ2+θ)] dθ
−A0∫
−τ1−τ2
Rft(t + θ) dθ + Rft
= − (In −A)x(t) −−τ1∫
−2τ1
x(t + θ) dθ
+ A−τ1−τ2∫
−2τ1−τ2
x(t + θ) dθ +
0∫−τ1
Rft(t + θ) dθ
+ A−τ1∫
−2τ1−τ2
x(t + θ) dθ −A2
−τ1−τ2∫−2τ1−2τ2
x(t + θ) dθ
−A0∫
−τ1−τ2
Rft(t + θ) dθ + Rft.
Consider a Lyapunov function candidate V (x) = xT Pfrx.Taking the derivative of V (x) along (13) gives
where we have used Lemma 4.1 and the facts that |xd| <δv and |xd| < δa to derive the inequality. Take p(s) = qfsfor some constant qf > 1. If V (x(t + θ)) < qfV (x(t)), for−2τ1 − 2τ2 ≤ θ ≤ 0, we have
V (x)≤xT[−(In −A)T Pfr − Pfr(In −A)
]x
+ τ1xT (Pfr + qfPfr)x
+ τ1xT
(PfrAP−1
fr AT Pfr + qfPfr
)x
+ (τ1+ τ2)xT(PfrAP−1
fr AT Pfr + qfPfr
)x
+ (τ1+ τ2)xT(PfrAAP−1
fr ATAT Pfr + qfPfr
)x
+ 2‖x‖ ‖Pfr‖[τ1(τ1+ τ2)δa + τ1τ2δv]
+ 2‖x‖ ‖PfrA‖[(τ1+ τ2)(τ1+ τ2)δa+(τ1+ τ2)τ2δv]
+ 2‖x‖ ‖Pfr‖[(τ1+ τ2)δa + τ2δv]
≤x(t)T Qftx(t) + af‖x‖
where Qft and af are defined as in Theorem 4.3. BecauseIn −A has all eigenvalues on the open right half-plane(Lemma 4.2), there exist τ1 and τ2 such that Qft < 0 if Pfr
is chosen such that (−In + A)T Pfr + Pfr(−In + A) < 0.Moreover, we have that λmin(−Qft) > 0. For 0 < κf < 1, if‖x‖ ≥ (af/κfλmin(−Qft)), we can obtain
Therefore, the uniformly ultimate boundedness of x followsfrom Lemma 3.2. Moreover, the ultimate bound is given byλmax(Pfr)af/λmin(Pfr)κfλmin(−Qft) by following a simi-lar analysis to that in [29, pp. 172–174]. �
Remark 4.7: Note that if τ1 = τ2 = 0, limt→∞ ‖x‖ = 0.Also, note that when τ1 and τ2 are larger, the bound will belarger.
D. First-Order Consensus Tracking With Partial Access to theVirtual Leader
Here, we assume that the time-varying reference states xd
and xd are available to only a portion of all agents and arebounded. We also assume that there exists only the commu-nication delay. Enlightened by [17], we propose the followingconsensus tracking algorithm with the communication delay:
ui =1∑n+1
j=1 aij
n+1∑j=1
aij {xj(t − τ2) − [xi(t) − xj(t − τ2)]} ,
i = 1, . . . , n (14)
where τ2 is the communication delay, aij , i = 1, . . . , n, j =1, . . . , n + 1, is the (i, j) entry of the adjacency matrix An+1,xn+1 ≡ xd, and xn+1 ≡ xd. Using (14), (3) can be written inthe matrix form as
x = Ax(t − τ2) − x(t) + Ax(t − τ2) + Rfft (15)
where A and x are defined as in Section IV-B, and Rfft =[xd(t − τ2) − xd(t)]1n − [xd(t) − xd(t − τ2)]1n.
MENG et al.: LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS 81
Theorem 4.4: If the fixed directed graph Gn+1 has a directedspanning tree, system (3) using (14) guarantees that all xi −xd are uniformly ultimately bounded no matter how large thecommunication delay is.
Proof: The proof follows from Lemma 3.3. First, it is easyto verify that ρ(A) < 1 based on the same analysis as that in[27], which means that the neutral operator Dxt = x −Ax(t −τ2) is stable. Consider a Lyapunov function candidate V (x) =xT x. It is easy to show that V (x) is positive definite. Taking thederivative of V (x) along (15) gives
If ‖Dxt‖ > ‖Rfft‖ (xd and xd are assumed bounded), we havethat V (Dxt) < 0. Therefore, the uniformly ultimate bounded-ness of x is guaranteed according to Lemma 3.3. �
Remark 4.8: From Theorem 4.4, it can be noted that thecommunication delay does not jeopardize the stability of thefirst-order system for the consensus tracking problem withpartial access to the virtual leader. However, with the increasein the communication delay, the tracking errors will increaseas well.
Remark 4.9: In real applications, the derivatives of theneighbors’ information states xj(t − τ2) can be calculated byusing numerical differentiation. For example, xj(t − τ2) canbe approximated by (xj(kT − τ2) − xj(kT − T − τ2))/T ,where T is the sampling period, and k is the discrete-time index.
V. SECOND-ORDER CASE WITH COMMUNICATION AND
INPUT DELAYS UNDER A DIRECTED NETWORK TOPOLOGY
Here, we model a group of agents with double-integratordynamics as
ri(t) = vi(t) vi(t) = ui(t), i = 1, . . . , n (16)
where ri, vi, and ui denote, respectively, the position, thevelocity, and the control input of the ith agent.
A. Second-Order Leaderless Consensus
The proposed leaderless consensus algorithm with both com-munication and input delays is given as
ui(t) = − 1∑nj=1 aij
n∑j=1
aij [ri(t − τ1) − rj(t − τ1 − τ2)]
− γc∑nj=1 aij
n∑j=1
aij [vi(t − τ1) − vj(t − τ1 − τ2)] ,
i = 1, . . . , n (17)
where τ1 and τ2 are, respectively, the input and communicationdelays, aij , i = 1, . . . , n, j = 1, . . . , n, is the (i, j) entry ofthe adjacency matrix An, and γc is a positive gain. Here, we
2According to Lemma 3.3, if we let p(s) = q2ff s for some constant qff >1,
we then know that p(V (Dxt)) > V (x(ξ)) for t − τ2 ≤ ξ ≤ t. However, thiscondition is not used in the proof because of the special expression of V .
also assume that every agent has a neighbor, which implies that∑nj=1 aij �= 0, ∀ i. The control objective here is to guarantee
that ri(t) → rj(t) and vi(t) → vj(t) as t → ∞ when thereexist both communication and input delays. Using (17), (16)can be written in the matrix form as[
r(t)v(t)
]=
[0n×n In
0n×n 0n×n
] [r(t)v(t)
]+
[0n×n 0n×n
−In −γcIn
]
×[
r(t − τ1)v(t − τ1)
]+
[0n×n 0n×n
A γcA
] [r(t − τ1 − τ2)v(t − τ1 − τ2)
]
where A is defined as in Section IV-A, r = [r1, . . . , rn]T , and
v = [v1, . . . , vn]T . Define rΔ= W−1r and v
Δ= W−1v, whereW is defined as in Section IV-A. Denote rn−1 and vn−1 as,respectively, the first n − 1 rows of r and v. Denote r2 and v2
as, respectively, the last row of r and v. System (17) can bedecoupled into the following:
2 Psc(A0 +A1 + A2)] < 0, where Psc is a symmetricpositive-definite matrix chosen properly such that(A0 + A1 + A2)T Psc + Psc(A0 + A1 + A2) < 0, andSsc and Hsc are arbitrary symmetric positive-definitematrices.
If the above conditions are satisfied, τ1 ∈ [0, τ1], and τ2 ∈[0, τ2], system (16) using (17) reaches consensus asymptoti-cally. Specifically, ri(t) → (pT v(0)/τ2) and vi(t) → 0, wherep is defined as in Theorem 4.1.
Proof: Similar to the analysis given in Section IV-A, wefirst prove that the stability of system (16) using (17) is guaran-teed if the three conditions in Theorem 5.1 are satisfied. Then,
82 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
we show that these three conditions are, indeed, satisfied whenGn has a directed spanning tree, every agent has a neighbor,and γc > γc. At last, the consensus equilibrium is explicitlypresented by using the final value theorem.
For system (18a), consider a Lyapunov function candidate
V(x(n−1)t
)=
⎡⎣xn−1(t) + A1
0∫−τ1
xn−1(t + θ) dθ
+ A2
0∫−τ1−τ2
xn−1(t + θ) dθ
⎤⎦
T
× Psc
⎡⎣xn−1(t) + A1
0∫−τ1
xn−1(t + θ) dθ
+ A2
0∫−τ1−τ2
xn−1(t + θ) dθ
⎤⎦
+
0∫−τ1
t∫t+θ
xn−1(ξ)T Sscxn−1(ξ) dξ dθ
+
0∫−τ1−τ2
t∫t+θ
xn−1(ξ)T Hscxn−1(ξ) dξ dθ.
Taking the derivative of V gives
V(x(n−1)t
)≤ xn−1(t)T Qscxn−1(t)
where Qsc is defined in Theorem 5.1. Thus, the stability ofsystem (18a) is guaranteed if conditions 2 and 3 are satisfiedby using Lemma 3.1.
For system (18b), define g(s) Δ= (γcs + 1)(e−τ1s −e−(τ1+τ2)s)/s2. By using the Nyquist stability criterion,we know that the stability of (18b) can be guaranteed if�{g(jω)} > −1, ∀ω ∈ (−∞,∞). Because
�{g(jω)} =−γc sin τ1ω + γc sin(τ1 + τ2)ω
ω
+− cos τ1ω + cos(τ1 + τ2)ω
ω2
=−γc sin τ1ω + γc sin(τ1 + τ2)ω
ω
−2 sin (2τ1+τ2)
2 ω sin τ22 ω
ω2
≥ − γcτ1 − γc(τ1 + τ2) −(2τ1 + τ2)τ2
2condition 1 in Theorem 5.1 guarantees the stability ofsystem (18b).
Next, we show that the three conditions in Theorem 5.1 are,indeed, satisfied if Gn has a directed spanning tree, every agenthas a neighbor, and γc > γc. It is straightforward to see thatthere exist τ1 and τ2 such that conditions 1 and 2 are satisfied.For condition 3, noting that
A0 + A1 + A2 =[0(n−1)×(n−1) In−1
−L −γcL
]
we know that the assumptions that Gn has a directed spanningtree, every agent has a neighbor, and γc > γc imply that all
eigenvalues of A0 + A1 + A2 are on the open left half-planeaccording to [30]. Thus, there always exists a Psc to guaranteethat (A0 + A1 + A2)T Psc + Psc(A0 + A1 + A2) < 0, whichimplies that condition 3 is satisfied.
For the consensus equilibrium, we know that the asymp-
totical stability of (18b) implies that
[r2(t)v2(t)
]→
[v2(0)/τ2
0
]as t → ∞, and the asymptotical stability of (18a) implies
that xn−1 → 0 as t → ∞. Thus, it follows that
[r(t)v(t)
]→[
(pT v(0)/τ2)1n
0n
]as t → ∞. �
Remark 5.1: Due to the existence of the communicationdelay, the final velocity is dampened to zero instead of a pos-sible nonzero constant as compared with the standard second-order consensus algorithm studied in [18]. Also, note that ifthere exists only the input delay, the final velocity is a possiblynonzero constant, and the final position is a ramp signal, whichare consistent with the results in [18].
Remark 5.2: Note that compared with the first-order casein Section IV-A, the second-order case requires more stringentconditions to guarantee stability, and the final consensus statesare different.
B. Second-Order Consensus Regulation With a ConstantFinal Velocity
Here, we assume that there exists a virtual leader, labeledas agent n + 1 with position rd and velocity vd. Here, weassume that vd is constant. The control objective here is toguarantee that all agents can track the virtual leader underlimited communication in the presence of delays. The proposedconsensus regulation algorithm is given as
ui = − 1∑n+1j=1 aij
n+1∑j=1
aij [ri(t − τ1) − rj(t − τ1 − τ2)]
− γr∑n+1j=1 aij
n+1∑j=1
aij [vi(t − τ1) − vj(t − τ1 − τ2)] ,
i = 1, . . . , n (19)
where τ1 and τ2 are, respectively, the input and communicationdelays, aij , i = 1, . . . , n, j = 1, . . . , n + 1, is the (i, j) entry ofthe adjacency matrix An+1, rn+1 ≡ rd, vn+1 ≡ vd, and γr is apositive gain. Note that if Gn+1 has a directed spanning tree,then it follows that
2 Psr(A0 + A1 + A2)] < 0, where Psr is asymmetric positive-definite matrix chosen properly suchthat (A0 + A1 + A2)T Psr + Psr(A0 + A1 + A2) < 0,and Ssr and Hsr are arbitrary symmetric positive-definitematrices.
In addition, if the above conditions are satisfied, τ1 ∈ [0, τ1],and τ2 ∈ [0, τ2], system (16) using (19) guarantees thatlimt→∞ r(t) → τ2vd(In −A)−11n and limt→∞ v(t) → 0n
asymptotically as t → ∞.Proof: Consider a Lyapunov function candidate
V (xt) =
⎡⎣x(t) + A1
0∫−τ1
x(t + θ) dθ
+ A2
0∫−τ1−τ2
x(t + θ) dθ
⎤⎦
T
× Psr
⎡⎣x(t) + A1
0∫−τ1
x(t + θ) dθ
+ A2
0∫−τ1−τ2
x(t + θ) dθ
⎤⎦
+
0∫−τ1
t∫t+θ
x(ξ)T Ssrx(ξ) dξ dθ
+
0∫−τ1−τ2
t∫t+θ
x(ξ)T Hsrx(ξ) dξ dθ.
Taking the derivative of V along (21) gives
V (xt) ≤ x(t)T Qsrx(t)
where Qsr is defined as in Theorem 5.2.By following a similar analysis to that in Section V-A,
we can prove the stability of (21) and the existence of τ1
and τ2 such that the two conditions in Theorem 5.3 are sat-
isfied. Since x(t) → 02n, as t → ∞, and M = [τ2vd[(In −A)−11n]T ,0T
n ]T , it follows that limt→∞ r(t) → τ2vd(In −A)−11n and limt→∞ v(t) → 0n asymptotically as t → ∞. �
Corollary 5.1: If vd = 0, we can get that limt→∞ ri(t) → rd
and limt→∞ vi(t) → 0 as t → ∞ given that the conditions inTheorem 5.2 are satisfied.
Remark 5.3: Note that different from the results in the first-order case in Section IV-B, the final positions of the followersmight not be equal in the second-order case. The final relativepositions of the followers are constant.
C. Second-Order Consensus Tracking With Full Access to theVirtual Leader
Here, the reference states rd, vd, and vd are assumed tobe time-varying, and vd is assumed to be available to allagents. The following consensus tracking algorithm with bothcommunication and input delays is proposed as
ui = vd(t − τ1 − τ2) −1∑n+1
j=1 aij
×n+1∑j=1
aij {[ri(t − τ1) − rj(t − τ1 − τ2)]
+ γt [vi(t − τ1) − vj(t − τ1 − τ2)]} ,
i = 1, 2 . . . , n (22)
where τ1 and τ2 are the input and communication delays,respectively, aij , i = 1, . . . , n, j = 1, . . . , n + 1, is the (i, j)entry of the adjacency matrix An+1, rn+1 ≡ rd(t), vn+1 ≡vd(t), and γt is a positive gain. We also assume that |vd| <δv, |vd| < δa, and |vd| < δa, where δv , δa, and δa are posi-tive constants. Using (22), (16) can be written in the matrixform as
where r, v, x, A, A0, A1, and A2 are defined as in Sec-
tion V-B, Rst =[0n
R1
], and R1 = −1n
∫ 0
−τ1−τ2vd(t + θ) dθ −
1n
∫ 0
−τ2vd(t + θ) dθ − γt1n
∫ −τ1
−τ1−τ2vd(t + θ) dθ by using the
Leibniz–Newton formula [19].Theorem 5.3: If the fixed directed graph Gn+1 has a
directed spanning tree and γt > γr, where γr is defined as inTheorem 5.2, there exist τ1 and τ2 such that Qst = (A0 +A1+A2)TPsr+Psr(A0+A1+A2)+ τ1(PsrA1A0P
−1sr AT
0 AT1
Psr + PsrA1A1P−1sr AT
1 AT1 Psr + PsrA1A2P
−1sr AT
2 AT1 Psr +
3qsPsr) + (τ1+τ2)(PsrA2A0P−1sr AT
0 AT2 Psr+PsrA2A1P
−1sr
AT1 AT
2 Psr + PsrA2A2P−1sr AT
2 AT2 Psr + 3qsPsr) < 0, where
Psr is the same matrix given in Theorem 5.2, and qs > 1. Inaddition, if Qst < 0, τ1 ∈ [0, τ1], and τ2 ∈ [0, τ2], system (16)using (22) guarantees that all ri − rd and vi − vd are uniformlyultimately bounded. In particular, the ultimate bound of xis given by λmax(Psr)as/λmin(Psr)κsλmin(−Qst), whereas = 2[‖Psr‖ + ‖PsrA1‖τ1+‖PsrA2‖(τ1+τ2)][(τ1+τ2)δa+τ2δv + γtτ2δa] and 0 < κs < 1.
84 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
Proof: Similar to the analysis given in Section IV-C, byusing the Leibniz–Newton formula [19], we transform (23) tothe following system:
d
dtx(t)= (A0 + A1 + A2)x(t) −A1A0
0∫−τ1
x(t + θ) dθ
−A21
−τ1∫−2τ1
x(t + θ) dθ −A1A2
−τ1−τ2∫−2τ1−τ2
x(t + θ) dθ
−A2A0
0∫−τ1−τ2
x(t+θ) dθ−A2A1
−τ1∫−2τ1−τ2
x(t+θ) dθ
−A22
−τ1−τ2∫−2τ1−2τ2
x(t + θ) dθ + Rst
−A1
0∫−τ1
Rst(t + θ) dθ −A2
0∫−τ1−τ2
Rst(t + θ) dθ.
Consider a Lyapunov function candidate V (x) = xT Psrx.Taking the derivative of V (x) along (23) gives
where we have used Lemma 4.1 and the facts that |vd| < δv ,|vd| < δa, and |vd| < δa to derive the inequality. Take p(s) =qss for some constant qs > 1. If V (x(t + θ)) ≤ qsV (x(t)) for−2τ1 − 2τ2 ≤ θ ≤ 0, by following a similar analysis to that inSection IV-C, we have that
V (x) ≤ x(t)T Qstx(t) + as‖x‖
where Qst and as are defined as in Theorem 5.3. It is easyto verify that there exist τ1 and τ2 such that Qst < 0 byfollowing a similar analysis to that in Section IV-C. More-over, we have that λmin(−Qst) > 0. For 0 < κs < 1, if ‖x‖ ≥as/κsλmin(−Qst), we can obtain that
The uniformly ultimate boundedness of x then follows fromLemma 3.2. Moreover, we can obtain that λmax(Psr)as/λmin(Psr)κsλmin(−Qst) is the ultimate bound of x by follow-ing a similar analysis to that in [29, pp. 172–174]. �
D. Second-Order Consensus Tracking With Partial Access tothe Virtual Leader
Here, we assume that the reference states rd, vd, and vd aretime-varying and available to only a portion of all agents. Wealso assume that the system is only influenced by the commu-nication delay. The proposed consensus tracking algorithm isgiven as
ui =1∑n+1
j=1 aij
n+1∑j=1
aij {vj(t − τ2) − [ri(t) − rj(t − τ2)] ,
− γft [vi(t) − vj(t − τ2)]} , i = 1, 2, . . . , n (24)
where τ2 is the communication delay, aij , i = 1, . . . , n, j =1, . . . , n + 1, is the (i, j) entry of the adjacency matrix An+1,rn+1 ≡ rd, vn+1 ≡ vd, vn+1 ≡ vd, and γft is a positive gain.Using (24), (16) can be written in the matrix form as
and r, v, A, and x are defined as in Section V-B.Theorem 5.4: If the fixed directed graph Gn+1 has a directed
spanning tree, and γft > γr, where γr is defined as in Theorem5.2, system (16) using (24) guarantees that all ri − rd and vi −vd are uniformly ultimately bounded if
λ > 2qsf ‖Psr(Af0Df + Af1)‖ + 2‖PsrAf1‖ (26)
MENG et al.: LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS 85
where λ = λmin[−(Af0 + Af1)T Psr − Psr(Af0 + Af1)],Psr is the same matrix given in Theorem 5.2, and qsf > 1.
Proof: First, it is easy to verify that ρ(Df ) < 1 based onthe same analysis as in [27], which means that the neutral op-erator Dxt = x − Dfx(t − τ2) is stable. Consider a Lyapunovfunction candidate V (x) = xT Psrx. Taking the derivative of Valong (25) gives
V (Dxt) = 2(Dxt)T Psr [Af0x + Af1x(t − τ2) + Rsft]= 2(Dxt)T Psr [Af0Dxt + Af0Dfx(t − τ2)
Letting f(s) = q2sfs for some constant qsf > 1, f(V (Dxt)) >
V (x(ξ)) for t − τ2 ≤ ξ ≤ t implies that q2sf (Dxt)T (Dxt) >
x(ξ)T x(ξ). It follows that x(t − τ2) < qsf (Dxt). Thus, itfollows that
V (Dxt) ≤ −λ‖Dxt‖2 + 2qsf‖Psr(Af0Df + Af1)‖ ‖Dxt‖2
+ 2‖PsrAf1‖ ‖Dxt‖2 + 2‖PsrRsft‖ ‖Dxt‖
where λ is defined in Theorem 5.4. Note here that the as-sumptions that Gn+1 has a directed spanning tree and γft >γfr guarantee that there exists A such that (26) is satisfied.Therefore, if λ > 2qsf‖Psr(Af0Df + Af1)‖ + 2‖PsrAf1‖,the uniformly ultimate boundedness of x can be achievedaccording to Lemma 3.3. �
Remark 5.4: Note that different from the first-order casewhere uniformly ultimate boundedness is guaranteed no mat-ter how large the communication delay is, a certain delay-independent condition has to be satisfied beforehand to en-sure the possibility of uniformly ultimate boundedness in thesecond-order case.
VI. SIMULATION
Here, we present simulation results to validate the theoreticalresults in Sections IV and V. We consider a group of six agents.For the leaderless consensus problem, the adjacency matrix An
Fig. 1. First-order cases. (a) Simulation results using (4). (b) Simulationresults using (9). (c) Simulation results using (12). (d) Simulation resultsusing (14).
86 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
For the first-order cases, the initial states are chosen asx(0) = [−1, 5, 7, 4, 6, 3]T . The input delay and the communi-cation delay are chosen, respectively, as τ1 = 0.1 s and τ2 =0.2 s. In the case of the first-order consensus regulation, welet the reference state be xd = 3.5. In the case of the first-order consensus tracking with full access to the virtual leader,we let the reference state be xd(t) = 3.5 − 4 cos(t/4). In thecase of the first-order consensus tracking with partial accessto the virtual leader, we let the reference state be xd(t) =3.5 − 4 cos(t/4) and the communication delay be τ2 = 0.2 s.
Fig. 1(a)–(d) shows the states of the agents for system (3)using, respectively, (4), (9), (12), and (14). It can be seen that forthe leaderless consensus and consensus regulation problems,there are no final tracking errors between the agents and thevirtual leader, while for the consensus tracking problem, thereexist bounded tracking errors between the agents and the virtualleader due to the existence of the delays and the fact that thevirtual leader is dynamic.
For the second-order cases, we choose r(0) = [−0.4, 0.5,0.7, 0.4, 1.2, 0.3]T and v(0) = [−0.1, 0.2, 0.7, 0.4,−0.1, 0.3]T
as the initial states. The input delay and the communicationdelay are chosen, respectively, as τ1 = 0.3 s and τ2 = 0.1 s.In the case of the second-order consensus regulation with azero final velocity, we let the reference states be rd = −0.2 andvd = 0. In the case of the second-order consensus regulationwith a nonzero constant final velocity, we let the referencestates be rd(t) = −0.2 + 0.1t and vd(t) = 0.1. In the case ofthe second-order consensus tracking with full access to thevirtual leader, we let the reference states be rd(t) = −0.2 +0.3t − 1.6 sin(t/4) and vd(t) = 0.3 − 0.4 cos(t/4). In the caseof the second-order tracking with partial access to the virtualleader, we let the reference states be rd(t) = −0.2 + 0.3t −1.6 sin(t/4) and vd(t) = 0.3 − 0.4 cos(t/4), and the commu-nication delay be τ2 = 0.1 s.
Fig. 2(a) shows the states ri and vi of system (16) using (17).It is interesting to notice that unlike the standard second-orderconsensus algorithm in [18], the final velocities are alwaysdampened to zero rather than a possibly nonzero constant.Fig. 2(b) and (c) shows, respectively, the states ri and vi ofsystem (16) using (19) when vd = 0 and vd = 0.1. It is worthnoticing that when vd is a nonzero constant, the final trackingerrors of all ri − rd approach constant (not necessary identical)values.
Fig. 2(d) and (e) shows the states ri and vi of system (16)using, respectively, (22) and (24). There exist bounded trackingerrors between the agents and the virtual leader due to theexistence of the delays and the fact that the virtual leader isdynamic.
VII. CONCLUSION
Leaderless consensus, consensus regulation, and consensustracking problems for both first-order and second-order inte-grators have been discussed under a directed network topologywith communication and input delays. By using decouplingtechniques, we have presented the stability conditions for theleaderless consensus problems. The consensus regulation prob-lems can be viewed as a direct extension of the leaderless
Fig. 2. Second-order cases. (a) Simulation results using (17). (b) Simulationresults using (19). (c) Simulation results using (19). (d) Simulation results using(22). (e) Simulation results using (24).
MENG et al.: LEADER-FOLLOWING CONSENSUS WITH COMMUNICATION AND INPUT DELAYS 87
consensus problems. In particular, the final velocities of theagents have been shown to be dampened to zero for the second-order leaderless consensus problem when there exists a com-munication delay. For the consensus tracking problems, theconditions to guarantee the uniformly ultimate boundedness ofthe tracking errors with full/partial access to the virtual leaderhave been presented. Finally, simulation results have been givento validate the theoretical results. Future works will includethe design of zero-error consensus tracking algorithms in thepresence of delays, the study on the case of consensus trackingalgorithms with partial access to the virtual leader when thereexist both communication and input delays, and the discussionon the influence of multiple time-varying delays.
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Ziyang Meng received the B.S. degree fromHuazhong University of Science and Technology,Wuhan, China, in 2006. He is currently working to-ward the Ph.D. degree in the Department of PrecisionInstruments and Mechanology, Tsinghua University,Beijing, China.
From 2008 to 2009, he was an exchange Ph.D. stu-dent supported by China Scholarship Council withthe Department of Electrical and Computer Engi-neering, Utah State University, Logan. His researchinterest focuses on cooperative control of distributed
multiagent systems and spacecraft attitude determination and control.
Wei Ren (S’01–M’04) received the B.S. degreein electrical engineering from Hohai University,Nanjing, China, in 1997, the M.S. degree in mecha-tronics from Tongji University, Shanghai, China, in2000, and the Ph.D. degree in electrical engineeringfrom Brigham Young University, Provo, UT, in 2004.
From October 2004 to July 2005, he was a Re-search Associate with the Department of AerospaceEngineering, University of Maryland, College Park.Since August 2005, he has been with the Departmentof Electrical and Computer Engineering, Utah State
University, Logan, where he is currently an Associate Professor. He is anauthor of the book Distributed Consensus in Multi-Vehicle Cooperative Control(Springer-Verlag, 2008). His research focuses on cooperative control of multi-vehicle systems, networked cyber-physical systems, and autonomous control ofunmanned vehicles.
Dr. Ren was a recipient of the National Science Foundation CAREER Awardin 2008. He is currently an Associate Editor for Systems and Control Letters andan Associate Editor on the IEEE Control Systems Society Conference EditorialBoard.
88 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 1, FEBRUARY 2011
Yongcan Cao (S’07) received the B.S. degree inelectrical engineering from Nanjing University ofAeronautics and Astronautics, Nanjing, China, in2003 and the M.S. degree in electrical engineer-ing from Shanghai Jiao Tong University, Shanghai,China, in 2006. He is currently working towardthe Ph.D. degree in the Department of Electricaland Computer Engineering, Utah State University,Logan.
His research interest focuses on cooperativecontrol and information consensus of multiagent
systems.
Zheng You received the B.S., M.S., and Ph.D. de-grees from Huazhong University of Science andTechnology, Wuhan, China, in 1985, 1987, and 1990,respectively.
In 1990, he joined the faculty of the Department ofPrecision Instruments and Mechanology, TsinghuaUniversity, Beijing, China, as an Assistant Professor,where he became an Associate Professor in 1992, aFull Professor in 1994, and was awarded speciallyappointed Professor for Chang Jiang Scholar by theMinistry of Education in 2001. He has published
more than 300 papers and 32 research reports. He is the holder of 12 Chineseinvention patents. His main research interests include micro–nano technologyand micro–nano satellite technology.
